As usual, straight continuous lines differentiate downwards with respect to x, and the straight dashed line with respect to the dashed balloon expression. So what you have here is two of the chain rule...

As usual, straight continuous lines differentiate downwards with respect to x, and the straight dashed line with respect to the dashed balloon expression. So what you have here is two of the chain rule...

Can you show me your work for the first one? I got a completely different answer.

Remember the quotient rule, where u and v are both quantities of x: $\displaystyle \frac{d}{dx} (\frac{u}{v}) = \frac{vu' - uv'}{v^2}$

Also in this case you will be applying the chain rule to both u and v since $\displaystyle u = (4x + 3)^7$ and $\displaystyle v = (5x - 2)^7$

Aug 9th 2009, 01:23 PM

tom@ballooncalculus

Whichever method you choose, it does no harm to know that the quotient rule is only the chain rule wrapped in the product rule, as depicted above. I don't say it's always an overwhelming advantage of the diagram but... please notice that the formula can land you with unnecessarily large powers in the denominator (14) which will probably want to be simplified down to what you already have in the diagram (8).

... although I must admit (not that one would necessarily care) that using the quotient rule inside the chain rule, while it makes for an easier ride with the formula, doesn't make one nice big picture. Nonetheless...